Question: Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{4x^2 - 32x + 64}{5x^3 - 45x^2 + 100x}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {4(x^2 - 8x + 16)} {5x(x^2 - 9x + 20)} $ $ p = \dfrac{4}{5x} \cdot \dfrac{x^2 - 8x + 16}{x^2 - 9x + 20} $ Next factor the numerator and denominator. $ p = \dfrac{4}{5x} \cdot \dfrac{(x - 4)(x - 4)}{(x - 4)(x - 5)}$ Assuming $x \neq 4$ , we can cancel the $x - 4$ $ p = \dfrac{4}{5x} \cdot \dfrac{x - 4}{x - 5}$ Therefore: $ p = \dfrac{ 4(x - 4)}{ 5x(x - 5)}$, $x \neq 4$